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A good reference for geodesic spaces (which includes a chapter dedicated to normed vector spaces) is Athanase Papadopoulos, Metric Spaces, Convexity, and Nonpositive Curvature (google books has a ...

It turns out this statement is not true without additional conditions on the space. See this counterexample for a closed convex set in as strictly convex Banach space which does not admit a metric ...

In general, $argmin$ is not continuous. Even on the real line, if I take $K$ to be two distinct points, say $K=\{-1,1\}$, then $argmin_{k\in K} d^2(x,k)$ is not continuous. This is why convexity is so ...

If we take the standard definition of a Jordan curve, that is a homeomorphic image of the unit sphere, then every Jordan curve in the Euclidean plane is compact. Given that every compact subset of a ...

Loosely speaking, if you are standing at the origin then there are only three directions you can travel. So the space of directions $S_o$ is only 3 points. Taking the product of the space of ...

Let $x,y,z,b \in X$ be given and assume that $\eta = \sup_{x\in X} d(x,Y) < \infty$. Fix $x' , y' , z' , b' \in Y$ such that $|x-x'|, |y-y'|$, etc. are all $\leq \eta$. (For a 'properly done' proof,...

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